Albert Einstein - Stafford Lectures (Meaning of Relativity)

In order to be able to do this, we must first solve the following problem. If we are given the Cartesian co-ordinates, x , and the time t , of an event relatively to one inertial system, K , how can we calculate the co-ordinates, x' , and the time, t' , of the same event relatively to an inertial system K' which move with uniform translation relatively to K ? In the pre-relativity physics this problem was solved by making unconsciously two hypotheses:--

1. Time is absolute; the time of an event, t' , relatively to K' is the same as the time relatively to K . If instantaneous signals could be sent to a distance, and if one knew that the state of motion of a clock had no influence on its rate, then this assumption would be physically validated. For then clocks, similar to one another, and regulated alike, could be distributed over the systems K and K' , at rest relatively to them, and their indications would be independent of the state of motion of the systems; the time of an event would then be given by the clock in its immediate neighbourhood.

2. Length is absolute; if an interval, at rest relatively to K , has a length s , then it has the same length s , relatively to a system K' which is in motion relatively to K .

if the axes of K and K' are parallel to each other, a simple calculation based on these two assumptions, gives the equations of transformation

This transformation is known as the "Galilean Transformation." Differentiating twice by the time, we get